In the context of the Semantic Web or semantic P2P systems, many ontologies may exist and be developed independently. Ontology alignments help integrating, mediating or simply reasoning with a system of networked ontologies. Though different formalisms have already been defined to reason with such systems, they do not consider ontology alignments as first class objects designed by third party ontology matching systems. Correspondences between ontologies are often asserted from an external point of view encompassing both ontologies. We propose a formalism, Integrated Distributed Description Logics (IDDL), which treats local knowledge (ontologies) and global knowledge (inter-ontology semantic relations, i.e. alignments) separately by distinguishing local interpretations and global interpretation. In this report, we identify relevant requirements for the semantics of such distributed systems. From this analysis, we argue that IDDL complies with these requirements. We then present a reasoning procedure for IDDL systems which uses local reasoners in a modular way. It proves that consistency of an IDDL system is decidable if consistency of the local logics is decidable and it provides an upper bound for the complexity of consistency checking.